Computing Joint Action Costs: Co-Actors Minimize the Aggregate Individual Costs in an Action Sequence

Successful performance in cooperative activities relies on efficient task distribution between co-actors. Previous research found that people often forgo individual efficiency in favor of co-efficiency (i.e., joint-cost minimization) when planning a joint action. The present study investigated the cost computations underlying co-efficient decisions. We report a series of experiments that tested the hypothesis that people compute the joint costs of a cooperative action sequence by summing the individual action costs of their co-actor and themselves. We independently manipulated the parameters quantifying individual and joint action costs and tested their effects on decision making by fitting and comparing Bayesian logistic regression models. Our hypothesis was confirmed: people weighed their own and their partner’s costs similarly to estimate the joint action costs as the sum of the two individual parameters. Participants minimized the aggregate cost to ensure co-efficiency. The results provide empirical support for behavioral economics and computational approaches that formalize cooperation as joint utility maximization based on a weighted sum of individual action costs.


. Participant exclusions
We excluded dyads (1) due to computing errors caused by equipment failure, which on occasion resulted in multiple disruptions during data collection; or (2) when the correlational structure of the experiment's parameters was not as intended, due to the design's stochastic nature; or (3) when participants chose the same object in every trial. Table S1.1 shows the total number of participants per experiment and the specific reasons for exclusion.  (13) 7 dyads were included in the final dataset who whose sessions were disrupted once, but successfully resumed (results from this sample were consistent with those form the rest of the group)

S.1.2.2. Experiments 2 and 3. Experiment 2 tested the hypothesis that action initiators
(Actor 1) plan their movements to minimize the summed aggregate action costs of the dyad's action sequence (Joint Cost Disparity) rather than to minimize their own individual costs (Self Cost Disparity). Experiment 3 probed the effect of Joint Cost Disparity against the individual costs of Actor 2 (Other Cost Disparity).
In both additional experiments, we applied the task from Experiment 1, and generated the layouts with the target objects' locations in the same way as in Experiment 1, with some important changes. We first sampled the individual -Self in Experiment 2, and Other in Experiment 3 -Cost Disparities for each trial from a triangular distribution with mode = 0 and limits provided by the maximum possible distance between an Actor's starting position and any target object (-265, 265 pixels). Then the parameters for Actor 2 (Other Disparity, Experiment 2) and Actor 1 (Self Disparity, Experiment 3), respectively, were drawn from a uniform distribution with limits set using the initially sampled Disparity parameter multiplied by -1.
Due to these sampling steps, the two actors' individual costs were negatively correlated with each other in both experiments (Fig. S1d, S1g), and the Joint Disparity defined by the two individual parameters' sum was independent from the Self Disparity (and positively correlated with Other Disparity, Figs. S2e-f) in Experiment 2, whereas it was independent from the Other Disparity in Experiment 3 (and positively correlated with Self Disparity, Figs. S1h-i). As in Experiment 1, every dyad in both experiments completed 200 trials (100 uniquely generated trials per participant) in a pseudo-random order.

S1.3. Description of the hierarchical model
We assumed that the trial-by-trial probability of choosing object A1 was Bernoulli distributed with parameter μi|s,k, where i indexes the trial, s indexes the participant and k indexes the experiment that the participants participated in (see Trial level in Fig. S2.). The value of this parameter depended on a logistic function of the focal cost parameter(s) of the model weighted by the participant's β coefficient/s, βSelf,s,k, βOther,s,k or βFairness,s,k (Subject level). The intercept was not estimated in the models, which is equivalent to assuming random decisions in the absence of any action cost disparities. The individual β coefficients were assumed to be normally distributed at the Experiment level around means µβSelf,k, µβOther,k, and µβFairness,k with standard deviations σOther,k, σSelf,k, and σOther,k, corresponding to the assumption that participants' individual weighing strategies are noisy versions of a shared group-level weighing pattern within an experiment. We included a Population level above the Experiment level with µβ and σβ values for each cost parameter's β coefficients. Each experiment's µSelf,k, µOther,k, µFairness,k, σSelf,k, σOther,k, and σFairness,k parameters were assumed to be sampled from the Population level, e.g. µβSelf,k ~ (µβSelf, σβSelf) and σβSelf,k ~ (0.0, 0.01). The priors for the Population level were set by hyperparameters µβSelf ~ (0, 5) and σβSelf ~ (0.0, 0.01) (similarly for the other disparity parameters), a distribution around a zero effect of cost disparity. The priors for the σβ parameters (and for the σβ,k, parameters one level below) were set to approximately match the ranges of posterior σβ estimates of the initial experiment-wise analyses 3 (Priors level). The same hyperpriors were used for all the predictors across all models.

S1.4. Technical information on the estimation process
We customized Bayesian data analysis scripts that are freely available online to accompany Kruschke (2015) 4 . Specifically, we adapted a multiple logistic regression model (Kruschke, 2015, p. 622) to predict a categorical dependent variable (object choice) in a hierarchical structure, which enabled the simultaneous estimation of individual, experiment-, and population-level β coefficient distributions.
All models were estimated using a Gibbs sampler in the runjags package (Denwood, 2016) in R (version 3.5.1). Three chains were initialized using fixed seeds of three random number generators for the reproducibility of results. At first, 1,000 adaptation steps and 10,000 burn-in steps were taken and discarded before reaching convergence between the three chains.
We kept 29,000 subsequent iterations for analysis, by thinning out every second step. Chain convergence was checked using Gelman and Rubin's (1992) convergence diagnostic, the potential scale reduction factor (PSRF). In most of the models, this factor's value was close to 1, i.e., chain convergence was satisfactory, and the full range of posterior distributions were explored. Although increasing the chain size would have ensured that all models' PSRF values be around 1, we had to compromise by capping the chain length at 29,000 iterations due to finite computational resources (to enable the calculation of WAIC and LOOIC measures for 3 N.B. Where comparison was possible, the experiment-level estimates did not qualitatively differ between the pooled analyses reported in the main text and in section S2 (Table S2.1), and the original, experiment-wise, analyses (reported in Table S3.1).
The original hyperpriors used for each experiment were µ ~ (0, 2) and σ ~ (0.0, 0.5). See section S.3 for details. model comparison, we had to estimate the log-likelihood at each trial, which placed considerable strain on our technical resources).
The data collected in the three experiments of the present study and the analysis scripts are available on the OSF site of the project: https://osf.io/r6mz3/?view_only=3f5fc782dac242adbe02bf3bc48158b0

S2. Results -Additional Experiment-wise Information
We report the experiment-level parameter estimates for the eight logistic regression models reported in the main text. First, Table S2.1 summarizes these, together with the population-level estimates and measures of model fit; then follows a detailed description of the results of the five main models.  To summarize, we found that in the case of the two single-predictor models, in 2 out of 3 experiments -when each of them was correlated with Joint Disparity -the disparities influenced decisions in the expected negative direction. The results of the estimations suggest that when each cost disparity parameter was de-correlated from the Joint Disparity of action sequences -i.e., Self Disparity in Experiment 2, and Other Disparity in Experiment 3 -, their effects were not as expected. Self Disparity by itself did not have an effect on choices (the 95% HDI included zero), whereas Other Disparity had an effect in the opposite direction than expected: when Other Disparity increased, the odds of an A1 choice also increased. This could possibly be due to an effect of self-cost minimization, because Other Disparity was negatively correlated with Self Disparity in Experiment 3.

S2.2. Model 3: Self and Other Disparities
In all three experiments, the experiment-level means (μβSelf,k and μβOther,k) of the βSelf and  suggests that with a one cm increase in the asymmetry in cost distribution between the two coactors, a 2.4% decrease in the odds of an object A1 choice over B1 was expected.
In Experiment 2, we found a small effect in the opposite direction: the 95% HDI of the posterior distribution of the μβFairness,2 estimates did not include zero, with a mode of 0.032 (95% In Experiment 2, we found an even larger overlap between the effect sizes of Self and Increasing each parameter by one cm resulted in expected decreases in the odds of an A1 choice by 33.0% (Self Disparity), 25.5% (Other Disparity), and 3.1% (Fairness).
Overall, the estimated weights on each parameter in the joint utility function, according to the combination model, were similar to one another across the three experiments. In .08]).

S3. Additional Experiment-wise Information: Separate analyses
We report in Table S3.1. the results of the original parameter estimations that we conducted on each experiment's data before pooling them together for the unified analyses. Since the designs of the three experiments differed in which pairs of cost disparities were de-correlated from one another, the models estimated also differed in the parameter combinations we used as predictors. Multiple-predictor models only included de-correlated parameter pairs; and in Experiments 2 and 3, we estimated only those single-predictor models of which the predictors were independent from Joint Disparity (i.e., in Experiment 2, we estimated only the Self Disparity model, in Experiment 3, only the Other Disparity model). In Experiment 1, although both individual cost disparity parameters were correlated with Joint Disparity, they were each tested as predictors in single-predictor models (Self Disparity only, Other Disparity only) to measure their predictive power against the combination model Self + Other Disparity.

S3.1. Description of the experiment-wise hierarchical models
The experiment-wise models were identical in structure to the described pooled data model, except for the removal of the experiment level.
We set the uninformed priors for this group-level distribution by vague hyperparameters (µ ~ (0, 2), σ ~ (0.0, 0.5)), a wide distribution around a zero effect of cost disparity. The same uninformed hyperprior was used for all cost disparities, expressing our prior expectation that participants would weigh the minimization of all costs equally (Priors level).

S4. Correlations between Perspective-Taking, Empathy, Liking a Co-actor and Behavior Data
It is possible that general abilities of perspective-taking and empathic concern in social interactions may prove useful in the computation of collective action costs in cooperative contexts. We report the results of exploratory correlational analyses (conducted on the pooled data) of the potential relationships between how much participants prioritized joint-cost minimization and their perspective-taking abilities and degree of empathy towards other people, as well as how much they liked their co-actors.
Following the object matching task and before being debriefed about the experiment, the participants responded to a short custom questionnaire on their perceived purpose of the study and how much they liked their partner ("How much did you like your co-player?"). Ratings of Liking the partner were obtained using a 7-point Likert scale (1 -Not at all, 7 -Very much).
Participants also completed the Perspective-Taking and Empathic Concern scales from the Davis Interpersonal Reactivity Index (Davis, 1980) as measures of perspective-taking and trait empathy. The maximum score on both scales was 28.
To operationalize the weight that participants placed on minimizing the joint costs of an action sequence, we used each participant's proportion of co-efficient choices out of the 100 trials they completed as the decision-making Actor 1 ("co-efficiency ratio"). The higher the value of this measure, the bigger the weight a participant placed on minimizing the joint costs of an action sequence.

S5. Figures showing participant-wise estimates for the best-fitting models
The figures below present the individual-level parameter estimates according to the bestfitting model. Fig. S3. shows the results of the analysis on the entire dataset, Fig. S4. on the data subset where the predictions of fairness and co-efficiency were dissociated, and Fig. S5.
on the first Block of 5 trials.    Measures of predictive accuracy and model fit (WAIC -Watanabe-Akaike Information Criterion, LOOIC -Leave-one-out Information Criterion, AUC -Area Under the Curve) of all of the logistic regression models mentioned in the main text. We include the 5 main models and the extended models addressing questions of learning and tit-for-tat decision-making.